Sarah Salem remembers well the moment Britain won the bid for the Olympic Games. She was watching television and in her excitement leaped up in the air. Salem, 63, suffered with polio as a child and more recently suffered a debilitating fall that largely confined her to her ninth-floor flat a mile or two from the Olympic Park in Hackney. "But I just knew that this was it, I had to be involved," she says. Salem had come to Britain in 1970 from Mauritius, and had always been "a bit of an Olympics nut".

It was Mary Peters who started her off in 1972. The news that the Games was coming to her neighbourhood was all the incentive she needed to change her life. She had worked for most of her life as a nurse and a social worker in Hackney, but after her fall suffered with depression and struggled to get out of her flat. "I had been like that for two years," she says. Then she heard about a programme called Personal Best, which aimed to involve those "furthest away from employment" in preparations for 2012, as volunteers.

Salem was accepted after badgering the tutor for weeks and then attended every class she could – how to manage crowds, health and safety, what to do in the event of a fire. "Nothing could stop me from going to the Olympics!" she says. Salem qualified as an Olympic Delivery Authority tour guide, and started showing groups around the park before it opened, "hospice groups, people from Stoke Mandeville suffering spinal injury, the IOC from the USA, the family of Jacques Rogge, the Olympic president. I mean: imagine!"

Salem is telling me this story in the press room of the Olympic Park basketball arena, where she worked all last week ("I also did an eight-week journalism course," she says, in passing). She is wearing the distinctive red and purple shirt of the 2012 volunteers with particular pride. The 70,000 volunteers were christened "Games makers" when they signed up for the job, after being chosen out of 400,000 applicants. Anyone who visited any of the Olympics venues last week and encountered the enthusiasm of people such as Sarah Salem would not think the phrase at all inaccurate.

Some volunteers have been assigned to particular specialist roles – as osteopaths or linguists or building equestrian fences, acting as trampoline "spotters" or fencing judges – but most are simply there to point out directions, manage bus queues, or shout a welcome to visitors. At the start of the week, there was just a little British nervousness about some of this (particularly apparent in the tentative use of megaphones). By about Tuesday, this had entirely disappeared. If there was a fear before this Olympics began that it would be a corporatised, soulless event, the effort and enthusiasm of the volunteers have filled it with a likeably amateur and properly human warmth.

People have volunteered for very different reasons. Many, such as Brett Akpan, a law student from Sheffield, who is marshalling at the basketball, are here just because, "it was always going to be a great place to be... and the outfit is like a hello – people just come up and talk, which is good". But many others, such as Sarah Salem, see volunteering as a genuine chance to change their lives, or to "give something back".

Sarah Salem's life has been transformed by working at the Olympics. Photograph: Antonio Olmos for the Observer Antonio Olmos/Observer

Jean Tomlin, head of human resources for the Olympic organising committee, says she was adamant from the start that volunteers would be as integral to the staffing of the Games as contracted workers, "a one-team approach". Their altruistic spirit is fundamental to the atmosphere of inclusion the Games have sought to foster. She takes inspiration from the 1948 "austerity Games" in London, "which first introduced widespread volunteering to the Olympic movement". The Personal Best programme is one example of how the organising committee has tried to extend the principle to harder-to-reach sectors.

In this, Locog has learned from and been supported by the initiative of the London boroughs that host the Olympic sites, in particular Newham, in which the main park is located. Newham council saw the Olympic bid as an opportunity to energise its volunteering programme. Nancy Whiskin has managed a hugely successful effort to engage local people in volunteering.

From 300 people in 2005 it now has 8,000 registered volunteers, most of whom have gone on to be "Games makers". "It has been a seven-year programme: 400,000 people volunteered to be part of the Games in total, so we knew our people had to be competitive to be engaged in this. We got them involved with a lot of high-profile sporting events, but also found opportunities in the council itself: they are helping in libraries, social services, children's services..."

Outside Stratford station near the Olympic park, some of the Newham Volunteers are assisting visitors to the area. Oliver McCarthy, 71, who worked for 40 years on the railways, explains his motivation: "I wanted to do something, not just sit. I have been retired eight years. Most of my colleagues were lucky if they got a year and a half of their pension. They didn't know what to do with themselves. The spark just went from them. You have to stay connected to people..."

Jessica Nyen, also part of the programme, came here as a refugee from Rwanda in 2000. "I came here to Newham with nothing," she says, "from terrible times in my country, and I was accepted. I wanted to do something to offer visitors to Newham a welcome, too."

"The idea that people will just do this without intervention, this 'big society' idea, is fundamentally flawed," Whiskin says. "You have to build on something. There has to be something linking people together." The Olympics has not only offered that opportunity, it has demonstrated the real enjoyment of volunteering: of being involved in something, and making connections. If there is to be a legacy of the Games, Whiskin and all the volunteers I speak to hope this will be part of it.

Sarah Salem certainly hopes so. "For the past month or so we have had this excitement, but also this trepidation," she says. "It is only just beginning but it is about to come to an end. I have been approached possibly to volunteer as a guide for the legacy tours of the park. I've made so many friends I wish it could go on for ever."